Did someone tell me this? I can’t recall. They ought to have, I’m sure. I’d be very surprised if Postnikov didn’t mention it when he first introduced those towers that today bear his name.

So, yes, every space $X$ "has a" Postnikov tower

$$ \xymatrix{

& \vdots \ar[d] \\

K(\pi_2 X,2) \ar[r] & [X]_2 \ar[d] & X \ar[ul]\ar[l]\ar[dl]\ar[ddl] \\

K(\pi_1 X,1) \ar[r] & [X]_1 \ar[d] & \\

& [X]_0 \rlap{{}\simeq \pi_0 X}

} $$

realizing $X$ as a limit (or, in HoTT, as a clear competitor to the limit…) of truncated spaces, in which each map $K\to [X]$ is the family of fibers for some base-point system…

Somewhat more special are the spaces which have a Postnikov *system*: these can be realized as limits of iterated fiber maps

$$ \xymatrix{ & \vdots \ar[d] \\

K(\pi_3 X,4) \ar@{<-}[r]_{k_3} & [X]_2 \ar[d] & X \ar[ul]\ar[l]\ar[dl]\ar[ddl] \\

K(\pi_2 X,3) \ar@{<-}[r]_{k_2} & [X]_1 \ar[d] & \\

K(\pi_1 X,2) & [X]_0 \rlap{{}\simeq \pi_0 X } \ar[l]^{k_1}

} $$ and the maps $k$ are called the $k$-invariants of the *system*.

It so happens that *simply-connected* spaces all have Postnikov systems. For another family of examples, it’s a complete triviality[*] already that *loopspaces* have Postnikov systems. The only observation needed is that the pullback squares

$$ \xymatrix{ \Omega [B]_n \ar[r]\ar[d] & K(\pi_{n+1} B, n+1) \ar[d] \ar[r] & * \ar[d] \\

* \ar[r] & [B]_{n+1} \ar[r] & [B]_n } $$ clarify the equivalences

$$ [\Omega B]_n \simeq \Omega [B]_{n+1} $$ as well as telling us what the $k$-invariants actually are, in particular that they are *fiber maps*. It’s just as trivial that a Postnikov system in which the $k$-invariants are fiber maps IS the Posntikov system *of a loopspace*.

Now, this still isn’t the thing I wanted to tell you about, but we’re at the right place to start, at last. Generalizing the preceding argument, it follows that the $k$-invariants of a double loopspace $X \simeq \Omega^2 W$ are *loops of fiber maps*, which will be the working lemma in

**Theorem** *Every connected retract $A$ of a double loop-space $\Omega^2 W$ is a loopspace with global commutator.*

**Remarks:** some folk would say “homotopy-abelian” instead of “with a global commutator”, but I’m afraid the traditional expression is much too vague. For the rest, we require “connected retract” simply because there’s too much noise involved in the more general case; you can figure out what should be said if we really want to say more.

**Proof**

We will argue by induction that, given a retraction

$$ A \to \Omega^2 W \to A $$ that the pieces $ [A]_n $ are loop-spaces, and then silently hope that the limit preserves this property. (Don’t worry: loop-spaces are limits, and limits commute with limits; the HoTT-worry is that $A$ *might not be* the limit of it’s Postnikov tower, but I don’t need to work in HoTT all the time, do I?)

To be sure, $[A]_0$ is a loop-space already, because it is a *point*. As well, $[A]_0 \to [\Omega^2 W]_0$ and $[\Omega^2 W]_0$ are loop-maps, being [without loss of generality] just homomorphisms of groups. That settles the base case. To induce up the tower, consider the diagram

$$ \xymatrix{

[A]_n \ar[d] & K(\pi_{n+1}A , n+2 ) \ar[d]\\

[\Omega^2 W]_n \ar[r]\ar[d] & K(\pi_{n+3} W, n+2 ) \ar[d] \\

[A]_n & K(\pi_{n+1} A, n+2) } $$ The remainder of the induction hypothesis is that both maps on the left are $\Omega$-maps; the vertical maps on the right are $\Omega$-maps, indeed $E_\infty$-maps; the map in the middle is a loop map by the working lemma, and so completing the diagram

$$ \xymatrix{

[A]_n \ar[d]\ar[r] & K(\pi_{n+1}A , n+2 ) \ar[d]\\

[\Omega^2 W]_n \ar[r]\ar[d] & K(\pi_{n+3} W, n+2 ) \ar[d] \\

[A]_n \ar[r] & K(\pi_{n+1} A, n+2) } $$ shows that $A$ has its $(n+1)$th $k$-invariant, which is, in particular, a loop map! It follows that the induced maps-of-fibers

$$ [A]_{n+1} \to [\Omega^2 W]_{n+1} \to [A]_{n+1} $$ are loop maps of loop spaces and also a section-retract pair, and so the induction is completed.

And, for the commutator: this is a homotopy filling the commuting triangle

$$ \xymatrix{ A^2 \ar[rr]^\tau \ar[dr]_m & & A^2 \ar[dl]^m \\

& A }$$ Exercise.